List by nation and unit.
It is estimated that anywhere from 400,000 to 1,000,000 non-German volunteers and conscripts served in the Waffen-SS, while millions more volunteered for service as various auxiliaries of the Axis war effort. Below is a popular estimate of numbers.

United States of America: 15 to 20
-SS-Standarte Kurt Eggers (plus various other divisions)

Plus the many Arab volunteers of the 'Free Arabia' Legion ('Freies Arabien'), whose members countries of origin ranged from Syria, Saudi Arabia, Egypt, Trans-Jordan, Palestine, Lebanon, Morocco, Tunisia, Algeria, Iran and Iraq, plus quite a few obscure others (like Senussi and Tuaregs and various desert Arab tribes and peoples). The numbers for the Free Arabia Legion were 20,000+.

In the closing days of the war, about 100 Arab soldiers under Major I.G. Schacht, in the 25th Parachute Regiment of the 9th Parachute Division, would even fight in defense of Berlin, right at its doorstep with their German comrades. Major Schacht wrote of their enormous courage, praising his Arab men for saving his life on numerous occasions.

The Russian people contributed greatly to the Axis cause as well. Modern research indicates that up to 1,500,000 Russian citizens volunteered. Almost 300,000 of those were found in combat units, while the rest were various auxiliaries and police formations.

Many Axis allies were to be found in occupied or liberated countries as well. Countries like Norway (under Vidkun Quisling's regime), Russia (under General Vlassov and others), Denmark, Ukraine, Greece, Macedonia, Lithuania, Latvia, Georgia, Azerbaijani... the list goes on and on.

There were also large numbers of supporters in unlikely places like America and Britain. The American Bund party rallies were so large they looked like they were held in Germany! And of course, many countries of South America were friends to the Axis countries, and even had National Socialist parties of their own.

[Above: Here is a postcard showing the shield and collar insignia of the many Waffen-SS divisions and their locale.]